Area bounded by the circle which is concentric with the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$and which passes through $\left(4,-\frac{9}{5}\right)$
the vertical chord common to both circle and ellipse on the positive side of x-axis is

$\text{ As eccentricity of ellipse }=\frac{4}{5}$

Coordinate of foci - (4,0), (-4, 0) 

$\Rightarrow \left(4,-\frac{9}{5}\right)\text{ is one of the end point of latus-rectum }$

Required area
$\begin{array}{l}=\frac{1}{2\pi }\times \pi \times \left(4^2+\frac{9^2}{5^2}\right)\times 2{\tan }^{-1}\left(\frac{9}{20}\right)-\text{ area of }\mathrm{\Delta }POQ\\ =\frac{481}{25}{\tan }^{-1}\left(\frac{9}{20}\right)-\frac{1}{2}\times 4\times \left(\frac{18}{5}\right)\\ =\frac{481}{25}{\tan }^{-1}\left(\frac{9}{20}\right)-\frac{36}{5}\end{array}$